

A194002


Numbers k that are the start of a sequence of 7 maximallysquarefree numbers.


3



1, 65, 137, 209, 217, 281, 353, 433, 641, 713, 785, 793, 857, 937, 1001, 1217, 1289, 1361, 1433, 1505, 1577, 1657, 1793, 1865, 1937, 2081, 2089, 2233, 2305, 2377, 2441, 2513, 2585, 2665, 2729, 2801, 2953, 3017, 3089, 3161, 3241, 3305, 3313, 3457, 3529, 3593
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OFFSET

1,2


COMMENTS

k, k+1, k+2, k+4, k+5, and k+6 are squarefree; k+3 is divisible by 4 but no higher power of 2 and no other prime squared.
All the terms are of the form 8*k + 1.
The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 14, 140, 1384, 13774, 137784, 1378053, 13779491, 137794128, 1377940943, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0137794... . (End)


LINKS



MATHEMATICA

sfQ[n_]:=Module[{c4=FactorInteger[n[[4]]], r=Drop[n, {4}]}, First[c4] == {2, 2} && Max[Transpose[Rest[c4]][[2]]]==1&&And@@SquareFreeQ/@r]; Join[{1}, Transpose[ Select[Partition[Range[2, 3600], 7, 1], sfQ]][[1]]] (* Harvey P. Dale, Nov 22 2011 *)


PROG

(PARI) ap(n)={forstep(k=1, n, 8,
if(issquarefree(k)&&issquarefree(k+1)&&issquarefree(k+2)&&
issquarefree((k+3)\2)&&
issquarefree(k+4)&&issquarefree(k+5)&&issquarefree(k+6),
print1(k", ")))}


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



